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arXiv · 2210.06414

Evolution Driven by the Infinity Fractional Laplacian

Félix del Teso·Jørgen Endal·Espen R. Jakobsen·Juan Luis Vázquez
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Abstract

We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland, Caffarelli and Figalli (2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions.

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Félix del Teso, Jørgen Endal, Espen R. Jakobsen, Juan Luis Vázquez. 2022-10-12. Evolution Driven by the Infinity Fractional Laplacian. https://doi.org/10.1007/s00526-023-02475-w

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